Unraveling the optical shape of snow

The reflection of sunlight off the snow is a major driver of the Earth’s climate. This reflection is governed by the shape and arrangement of ice crystals at the micrometer scale, called snow microstructure. However, snow optical models overlook the complexity of this microstructure by using simple shapes, and mainly spheres. The use of these various shapes leads to large uncertainties in climate modeling, which could reach 1.2 K in global air temperature. Here, we accurately simulate light propagation in three-dimensional images of natural snow at the micrometer scale, revealing the optical shape of snow. This optical shape is neither spherical nor close to the other idealized shapes commonly used in models. Instead, it more closely approximates a collection of convex particles without symmetry. Besides providing a more realistic representation of snow in the visible and near-infrared spectral region (400 to 1400 nm), this breakthrough can be directly used in climate models, reducing by 3 the uncertainties in global air temperature related to the optical shape of snow.


Supplementary Figures
Supplementary Figure 1 a b Fig. 1 Estimation of the optical shape parameters with the macroscopic method for the I23 (RG) sample. a Model estimations of the absorption enhancement parameter B represented as probability distributions, with the lines in each violin plot corresponding to the extrema, the mean, and the 10th and 90th-percentile of the resulting distributions. The geometric method estimation is also displayed. b Model estimations of the geometric asymmetry parameter g G represented as probability distributions.
Supplementary Figure 2 a b Fig. 2 Spectral variations of the optical shape parameters. a Absorption enhancement parameter B. b Geometric asymmetry parameter g G . In all panels, both retrieval methods are shown, and all snow microstructure images are considered. The Malinka estimations for the two-phase random medium are included. The lower and upper limits of the envelope wrapping the median value represent the 10th and the 90th-percentile of the estimates for each wavelength. Note that the B values for the two-phase random medium are virtually equivalent to those estimated with the geometric method and equal to n 2 . Note also that the macroscopic method is less accurate below 600 nm and above 1200 nm (see Method limitations). This is the reason why the envelope is more transparent for the shortest and longest wavelengths.  and flux profile (I(z)) (bottom) at λ = 900 nm of three different snow samples. SSA is the specific surface area, i.e. the total surface area of the air-ice interface per unit of mass, and ρ is the sample density. Spherical and fractal scenarios are computed using the AART theory [1][2][3][4] with the corresponding values of the optical shape parameters B and g G . Figure 4 a b c d f e

Supplementary Tables
Supplementary Table 1   Table 1 Description of the 3D snow microstructure images dataset. Resolution corresponds to the X-ray tomography imaging pixel size, and snow properties (density and specific surface area (SSA)) were computed directly over the mesh with the trimesh Python package [5]. The snow samples were either collected in the field (French Alps) or come from controlled cold-room experiments. In the latter case, the initial sample was most of the time recent alpine snow. More details on the snow sampling and characterization are available at each of the correspondent studies. Supplementary Table 2   Table 2 Optical shape parameters of geometric shapes. Values of B and g G for a diverse set of geometric shapes, computed with ray-tracing models in previous studies. The aspect ratio for hexagonal plates, cuboids and cylinders is the ratio of the height to, respectively, the length of the hexagonal side, the length of the square side and the radius of the circle. For the spheroid, it is the ratio of the largest semi-axis to the shortest one.

Supplementary Methods
Supplementary Methods 1: B and g G for the two-phase random medium For the sake of completeness, we show here the expressions of the optical shape parameters B and g G for the two-phase random medium, which are actually derived in [13]. Considering that in such medium, along a straight line, the positions of consecutive air-ice interfaces follows a Markov process, it was shown that B = n 2 in the limit of low absorption (see Eqs. 9 and 25 in [13]). The geometric asymmetry parameter g G is analytically derived in the same study (Eq. 60) as: where ω 0 is the single-scattering albedo (named photon survival probability in [13] -Eq. 56), L(α) is the Laplace transform of the chord length distribution of the medium (Eq. 21), and where r 1 , t 1 and r in 1 are given in an analytic way in Eqs. 29, 42 and 49 in [13].
The value of these parameters for the two-phase random medium are wavelength-dependent via the ice refractive index n, and their median values across the visible and NIR spectral range (400 -1400 nm) are B = 1.69 and g G = 0.67.

Supplementary Methods 2: Analytical convex shape generation
The analytical convex shape explored in this study is generated with the following parametric equations: where x ′ and y ′ are defined as: x ′ = cos ϕ · sin θ y ′ = sin ϕ · sin θ with the polar angle θ ∈ [0, π] and the azimuthal angle ϕ ∈ [0, 2π].

Supplementary Methods 3: Fresnel's law of reflectance
When the ray path intersects an ice-air interface, a decision between reflection and refraction (and therefore change of direction) may be done. This choice is random and depends mainly on a probabilistic interpretation of the Fresnel coefficients, defined by: where n i is the refractive index of media 1 and 2 (i.e. i = 1, 2) and θ i is the angle of incidence between the incoming ray v i and the normal vector of the interface v n (oriented towards the medium 1). The total reflected energy is then computed as: and therefore the transmitted energy would simply be T = 1−R. Here, this deterministic interpretation is treated with a Monte Carlo approach, where a random number is drawn and compared to R. If inferior, the ray encounters a reflection, otherwise a refraction. In both cases, the ray carries all the incident energy. In case of reflection, the ray outgoing direction v o is defined by: and in case of refraction: